Tony Huynh

Tony Huynh, Eun Jung Kim, Sang-il Oum et al.

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.

J. Pascal Gollin, Maximilian Gorsky, Meike Hatzel et al.

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $ψ$ on a surface $Σ$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(ψ)$ of $ψ$, and let $R(G,ψ)$ be the radial graph of $(G, ψ)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(ψ)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $ψ$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(ψ) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,ψ)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.